What is the integral of (3^{(2-x)}+(x+1)^2e^{(1-x)}) ?

The integral of (3^{(2-x)}+(x+1)^2e^{(1-x)}) is -(3^{2-x})/(ln(3))+e(-e^{-x}x^2-4e^{-x}x-5e^{-x})+C

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integral of (3^{(2-x)}+(x+1)^2e^{(1-x)})

\int (3^{(2 - x)} + (x + 1)^{2} e^{(1 - x)}) d x

- \frac{3 ^{2 - x}}{ln ( 3 )} + e (- e^{- x} x^{2} - 4 e^{- x} x - 5 e^{- x}) + C

Solution steps

\int 3^{2 - x} + (x + 1)^{2} e^{1 - x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3^{2 - x} d x + \int (x + 1)^{2} e^{1 - x} d x

\int 3^{2 - x} d x = - \frac{3 ^{2 - x}}{ln ( 3 )}

\int (x + 1)^{2} e^{1 - x} d x = e (- e^{- x} x^{2} - 4 e^{- x} x - 5 e^{- x})

= - \frac{3 ^{2 - x}}{ln ( 3 )} + e (- e^{- x} x^{2} - 4 e^{- x} x - 5 e^{- x})

Add a constant to the solution

= - \frac{3 ^{2 - x}}{ln ( 3 )} + e (- e^{- x} x^{2} - 4 e^{- x} x - 5 e^{- x}) + C

What is the integral of (3^{(2-x)}+(x+1)^2e^{(1-x)}) ?
The integral of (3^{(2-x)}+(x+1)^2e^{(1-x)}) is -(3^{2-x})/(ln(3))+e(-e^{-x}x^2-4e^{-x}x-5e^{-x})+C